Integral Test

Definition

Integral Test

Let $f(x)$ be continuous, non-increasing, and non-negative on $[1, +\infty)$ with $a_n = f(n)$ . Then the series $\sum_{n=1}^{\infty} a_n$ and the integral $\int_1^{+\infty} f(x) dx$ share the same convergence behavior.

符号说明

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$\sum$	Greek letter	Sigma	Summation symbol, representing series
$\infty$	Mathematical symbol	Infinity	Represents infinite series, infinite number of terms
$\int$	Mathematical symbol	Integral symbol	Used for definite or improper integrals

Formula

Integral Test

The series $\sum_{n=1}^{\infty} a_n$ and the integral $\int_1^{+\infty} f(x) dx$ converge or diverge together, where $a_n = f(n)$ and $f(x)$ is continuous, non-increasing, and non-negative on $[1, +\infty)$ .

Applicable Scenarios

The general term can be represented by a continuous function.
Other convergence tests are difficult to apply.

Example

Example 1

Determine whether the series $\sum_{n=1}^{\infty} \frac{1}{n\ln n}$ converges.

Solution: Let $f(x) = \frac{1}{x\ln x}$ so that $a_n = f(n)$ .

Evaluate the integral: $\int_2^{+\infty} \frac{1}{x\ln x} dx = \int_2^{+\infty} \frac{1}{\ln x} d(\ln x) = \ln(\ln x) \big|_2^{+\infty} = +\infty$

The integral diverges, so the series diverges as well.

Exercises

Exercise 1

Determine whether the series $\sum_{n=1}^{\infty} \frac{1}{n\ln n}$ converges.

Reference Answer (2 个标签)

series convergence integral test

Solution Idea： Apply the integral test to compare the series and the corresponding integral.

Detailed Steps：

Let $f(x) = \frac{1}{x\ln x}$ so that $a_n = f(n)$ .
Evaluate the integral: $\int_2^{+\infty} \frac{1}{x\ln x} dx = \int_2^{+\infty} \frac{1}{\ln x} d(\ln x) = \ln(\ln x) \big|_2^{+\infty} = +\infty$ .
The integral diverges, hence the series diverges.

Answer： The series diverges (divergence).

Summary

Symbols in This Article

Symbol	Type	Pronunciation / Note	Meaning in This Article
$f(x)$	Mathematical symbol	Function	A continuous function
$a_n$	Mathematical symbol	General term	The $n$ th term of the series
$\ln$	Mathematical symbol	Natural logarithm	The natural logarithm function

Chinese-English Glossary

中文术语	英文术语	音标	说明
积分判别法	integral test	/ˈɪntɪɡrəl test/	通过积分判断级数收敛性的方法
收敛	convergence	/kənˈvɜːdʒəns/	级数部分和序列有有限极限
发散	divergence	/daɪˈvɜːdʒəns/	级数部分和序列无有限极限
连续函数	continuous function	/kənˈtɪnjʊəs ˈfʌŋkʃən/	在定义域内连续的函数
单调递减	monotonically decreasing	/ˌmɒnəˈtɒnɪkli dɪˈkriːsɪŋ/	函数值随自变量增大而减小

PreviousRoot TestLearn the definition, usage, and applications of the root test (Cauchy's test). NextLeibniz TestLearn the definition, usage, and applications of the Leibniz test.

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